Generalized Thue-Morse sequences of squares

We consider compact group generalizations T(n) of the Thue-Morse sequence and prove that the subsequence T(n ²) is uniformly distributed with respect to a measure gv that is absolutely continuous with respect to the Haar measure. The proof is based on a proper generalization of the Fourier based method of Mauduit and Rivat in their study of the sum-of-digits function of squares to group representations.